Problem on Helly's theorem

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Let F be a finite family of segments in R such that among any n of them there are two mutually disjoint. We have to prove that it is possible to divide F into n− 1 family such that any two segments in one family are mutually disjoint.

So I have tried solving it in this manner as follows: enter image description here

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Please let me know if its a correct approach or am I missing out something.